COMPUTER-GENERATED HOLOGRAPHIC DISPLAY SYSTEM

Abstract

Computer-generated holographic display systems are described with improved image quality when used with illumination sources other than single mode lasers, such as broader emission light sources including LEDs and multi-mode lasers.

Claims

1. A computer-generated holographic display system having an angular resolution and an angular field of view at a viewing position, the computer-generated holographic display system comprising: an illumination system comprising an LED or multimode laser; an SLM illuminated by the illumination system; and an optical system configured to reimage the SLM at a predetermined distance from the viewing position; wherein: the illumination system has an etendue at the SLM which is less than or equal to a product of the angular resolution at the viewing position and the angular field of view at the viewing position divided by a maximum focal power of a virtual image point with respect to the reimaged SLM; and the angular resolution is less than about 1 mrad.

2. The computer-generated holographic display system according to claim 1, wherein the viewing position is a position of a viewer's pupil in use.

3. The computer-generated holographic display system according to claim 1, wherein the illumination system has an etendue greater than 2% of a product of the angular resolution and the angular field of view divided by a maximum focal power of a virtual image point with respect to the reimaged SLM.

4. The computer-generated holographic display system according to claim 1, wherein the illumination system comprises an angular filter which at least partially sets the source etendue of the illumination system.

5. The computer-generated holographic display system according to claim 1, wherein the illumination system comprises a spatial filter which at least partially sets the source etendue of the illumination system.

6. The computer-generated holographic display system according to claim 1, wherein the predetermined distance at which the SLM is reimaged is greater than a minimum distance of a virtual image point from the viewing position.

7. The computer-generated holographic display system according to claim 1 wherein the maximum focal power corresponds to a virtual image point less than or equal to about 1 m from the viewing position.

8. The computer-generated holographic display system according to claim 1, wherein the maximum focal power corresponds to a virtual image point approximately 0.25 m from the viewing position.

9. The computer-generated holographic display system according to claim 1, wherein the maximum focal power is less than or equal to about 3 dioptres.

10. The computer-generated holographic display system according to claim 1, wherein the angular resolution is greater than about 0.15 mrad.

11. The computer-generated holographic display system according to claim 1, wherein: the computer-generated holographic display system has a limiting aperture width; the illumination system has a spectral bandwidth and a nominal wavelength; and the spectral bandwidth divided by the nominal wavelength is less than or equal to the angular resolution divided by the product of the maximum focal power and the limiting aperture width.

12. The computer-generated holographic display system according to claim 11, wherein the limiting aperture width is less than about 7 mm.

13. The computer-generated holographic display system according to claim 1, wherein: the illumination system has a spectral bandwidth and a nominal wavelength; and the spectral bandwidth divided by the nominal wavelength is less than or equal to the angular resolution divided by the product of the maximum focal power and 7 mm.

14. A computer-generated holographic display system having an angular resolution at a viewing position and a limiting aperture width, the computer-generated holographic display system comprising: an illumination system comprising an LED or multi-mode laser and having a spectral bandwidth and a nominal wavelength; an SLM illuminated by the illumination system; and an optical system configured to reimage the SLM at a predetermined distance from the viewing position; wherein: the spectral bandwidth divided by the nominal wavelength is less than or equal to the angular resolution divided by the product of the limiting aperture width and a maximum focal power of a virtual image point with respect to the reimaged SLM; and the angular resolution is less than about 1 mrad.

15. A computer-generated holographic display system having an angular resolution at a viewing position, the computer-generated holographic display system comprising: an illumination system comprising an LED or multi-mode laser and having a spectral bandwidth and a nominal wavelength; an SLM illuminated by the illumination system; and an optical system configured to reimage the SLM at a predetermined distance from the viewing position; wherein: the spectral bandwidth divided by the nominal wavelength is less than or equal to the angular resolution divided by the product of a maximum focal power of a virtual image point with respect to the reimaged SLM and 7 mm; and the angular resolution is less than about 1 mrad.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0047] FIG. 1 shows an example viewing scheme for a CGH display, illustrating the effects of source etendue, where an SLM is reimaged some distance from a viewer;

[0048] FIG. 2 shows an example viewing scheme for a CGH display illustrating effects of spectral bandwidth of the source, where an SLM is reimaged some distance from a viewer;

[0049] FIG. 3 shows an example CGH display system in which the principles of this disclosure can be implemented.

[0050] FIG. 4 shows an example of how an angular subtense contributes to the point spread function of the display system;

[0051] FIG. 5 shows an example of how spectral bandwidth contributes to the point spread function of the display system;

[0052] FIG. 6 shows the effect of a relay lens in the CGH display system of FIG. 3; and

[0053] FIG. 7 is a flow chart of an example method to determine illumination system characteristics.

DETAILED DESCRIPTION

[0054] The present disclosure considers contributions to a Point Spread Function (PSF) at a viewing position to allow the impact of broad bandwidth sources (such as in terms of spectral bandwidth and etendue) on image resolution of a computer-generated hologram display (hereafter also referred to as a holographic display) to be understood. From this analysis various relations can be determined to improve the design of CGH displays.

[0055] FIG. 1 shows a generic viewing scheme for a CGH display, comprising: an image of an SLM, 102; a virtual image point that is viewed as part of the displayed image, the point being in a plane 104; and a viewer, viewing from a plane 106. The image of the SLM is at some distance, d.sub.SLM, from the viewer, shown as 108; a virtual image point is at distance, d.sub.obj, from the viewer shown as 110. The distance between the re-imaged SLM and a virtual object, d is also shown, shown as 112.

[0056] The location of the virtual object is described as being some focal power away from the plane of the SLM, where the focal power D is defined by D=(d.sub.SLMd.sub.obj)/(d.sub.SLM d.sub.obj), with units of dioptres.

[0057] Note that in FIG. 1 the image of the SLM is shown as being at a finite distance in front of the viewer for simplicity. However, the SLM may be reimaged at infinity, or beyond infinity, meaning the position of the virtual image has wrapped around and is behind the viewer and is d.sub.SLM negative. In cases such as these the following analysis is still applicable.

[0058] Due to the extended nature of the source, light at the image of the SLM has some angular subtense, .sub.SLM, shown as 114. This results in the PSF in the plane of the object having some width, w.sub.spatial, shown as 116. Working in the small angle approximation:

w.sub.spatiald.Math..sub.SLM.

[0059] This results in a PSF as seen by the viewer having some angular width .sub.spatial, shown as 118. .sub.spatial represents the contribution to the overall angular psf due to the limited spatial coherence of the source.

[0060] Also shown is the width of the image of an SLM, w.sub.SLM, shown as 120, and the angular field-of-view, .sub.fov, shown as 122. In other words, w.sub.SLM is the apparent width of the SLM as viewed from plane 106, such as the width of an image of the SLM when viewing the SLM through an optical system positioned between the SLM and the plane 106.

[0061] Conservation of etendue from light source to SLM means that the product of .sub.SLM and w.sub.SLM, which we denote G, is the same as the product of the angular width of the source, .sub.source, with the width of the source, w.sub.source. Put another way:

[00004]$w_{\mathrm{SLM}}=\left({}_{SLM}.w_{\mathrm{source}}\right)/{}_{source}$

Note that the extent of the source here is defined as being the extent of light incident on the SLMe.g., if the source is spatially or angularly filtered then those dimensions define the extent of the source. For simplicity, in this description G denotes etendue in a single dimension (i.e., a length multiplied by an angle). The same analysis may be followed in two dimensions, considering the extent of the source in x and y, where G is then defined by an area multiplied by a solid angle.

[0062] We now proceed to show that the etendue of the source, G, can be specified in terms of .sub.spatial, .sub.fov and D: i.e., for a display specified to have a given angular resolution, field of view, and where objects are displayed within some focal power of the SLM, the maximum permissible source etendue can be specified.

[0063] Working in the small-angle approximation (sin xtan xx), we can write

[00005]${}_{\mathrm{spatial}}=w_{\mathrm{spatial}}/d_{\mathrm{obj}}$$\mathrm{and}$$w_{\mathrm{spatial}}=d.{}_{\mathrm{SLM}}=\left(d_{\mathrm{SLM}}-d_{\mathrm{obj}}\right){}_{\mathrm{SLM}}$$\mathrm{also}$${}_{\mathrm{SLM}}=G/w_{\mathrm{SLM}}=G/\left(d_{SLM}.{}_{\mathrm{fov}}\right).$

[0064] Putting this together we obtain:

[00006]${}_{\mathrm{spatial}}=\left(d_{\mathrm{SLM}}-d_{\mathrm{obj}}\right){}_{\mathrm{SLM}}/d_{\mathrm{obj}}=G\left(d_{\mathrm{SLM}}-d_{\mathrm{obj}}\right)/\left(d_{\mathrm{obj}}.d_{\mathrm{SLM}}.{}_{\mathrm{fov}}\right)=G.D/{}_{\mathrm{fov}}$

So, expressing this in terms of source etendue for a given .sub.spatial we obtain:

[00007]$\begin{array}{cc}G={}_{\mathrm{spatial}}{}_{\mathrm{fov}}/D& \left(1\right)\end{array}$

[0065] The analysis for .sub.temporal, the contribution to angular PSF due to a finite source bandwidth, , is similar. FIG. 2 shows a generic viewing scheme for a CGH display. As per the temporal bandwidth analysis, the display comprises: an image of an SLM, 202; a virtual image point that is viewed as part of the displayed image, the point being in a plane 204; and a viewer, viewing from a plane 206. The image of the SLM is at some distance, d.sub.SLM, from the viewer, shown as 208; a virtual image point is at distance from the viewer, d.sub.obj, shown as 210. The distance between the re-imaged SLM and a virtual object, d is also shown, shown as 212.

[0066] Light rays 214, drawn as solid lines, for an image point a distance d.sub.obj, from the viewer, displayed at the nominal design wavelength, . Light rays 214 converge on the plane 204. A second set of light rays 216, drawn as dashed lines, correspond to light receiving the same phase modulation at the SLM, but having a different wavelength, =+. Due to the difference in wavelength, the rays 216 converge on a different plane shown as 218. Note that the scales are adjusted for clarity; in reality the distance 210 of the virtual image point from the viewer is likely to be at least 100 times greater than the distance 220, meaning that the rays 214 and 216 are close to being congruent.

[0067] The extent of the rays at the design wavelength are defined by the width of a limiting aperture, w.sub.eyebox, shown as 220. This aperture is likely to be an image of a spatial filter in the display apparatus, rather than a physical aperture in the plane of the eye. However, in the case where the eyebox (positions in which the image can be viewed) is larger than the viewer's pupil, the width of the pupil may be considered as the limiting aperture.

[0068] In the plane of the object, 204, the rays 216 define some width, w.sub.temporal, shown as 222. This in turn defines .sub.temporal, the contribution to the overall angular PSF due to the limited temporal coherence of the source, shown as 224.

[0069] 226 and 228 show construction lines for calculating .sub.temporal. 226 is a line from the image of the SLM to the centre of the eyebox, and 228 is a ray passing through the image point in the plane 204. 226 represents a ray that has zero focal power from the SLM (i.e., an image point in the plane of the SLM), and 228 shows a ray starting from the same point at the SLM but deflected to contribute to the object point in the plane 204. The angular deflection at the design wavelength is shown as 230. At the different wavelength , the deflection angle , shown as 232, is altered. The difference in deflection angle due to spectral bandwidth, =, is shown as 234. For small angles, diffraction angle is proportional to wavelength, so =/. It can also be seen that in the limit of 214 and 216 approaching congruence, =w.sub.eyebox/(2d.sub.SLM), so =w.sub.eyebox /(2d.sub.SLM)

[0070] We can now proceed to calculate .sub.temporal. Once again working in the small-angle approximation, we can write:

[00008]${}_{\mathrm{temporal}}=w_{\mathrm{temporal}}/d_{\mathrm{obj}}$

[0071] Substituting in our expression for from above:

[00009]$w_{\mathrm{temporal}}=2d_{\mathrm{obj}}=w_{\mathrm{eyebox}}{d}_{\mathrm{obj}}/\left(d_{\mathrm{SLM}}\right)$$\mathrm{So}$${}_{\mathrm{temporal}}=w_{\mathrm{eyebox}}{d}_{\mathrm{obj}}/\left(d_{\mathrm{obj}}{d}_{\mathrm{SLM}}\right)=w_{\mathrm{eyebox}}D/$

[0072] Expressing this in terms of spectral bandwidth for a given .sub.temporal we obtain:

[00010]$\begin{array}{cc}/={}_{\mathrm{temporal}}/\left(D.w_{\mathrm{eyebox}}\right)& \left(2\right)\end{array}$

Example System Designs

[0073] The above analysis is conducted without specific reference to SLM properties such as number of pixels or pixel pitch. This allows for powerful and easy analysis of designs based on user-facing specifications, such as angular resolution, field-of-view and eyebox size (the dimensions of the area where the image can be viewed).

[0074] In an example, image points are within some maximal absolute focal power of the image of the SLM, D.sub.max, and additionally .sub.spatial and .sub.temporal are both required to be smaller than some target angular resolution, .sub.resolution. .sub.resolution can be thought of as the largest numerical value of angular resolutionthat is to say, the worst angular resolutionthat meets the design requirements for the display. In a display, .sub.resolution corresponds to the resolution that can be measured in use at the viewing position, for example.

[0075] This lets us rephrase (1) and (2) as the inequalities:

[00011]$\begin{array}{cc}G{}_{\mathrm{resolution}}{}_{\mathrm{fov}}/D_{\max }& \left(3\right)\end{array}$$\mathrm{and}$$\begin{array}{cc}/{}_{\mathrm{resolution}}/\left(D_{\max }{w}_{\mathrm{eyebox}}\right)& \left(4\right)\end{array}$

Examples 1 to 3Specifying an Illumination Source

[0076] The following worked examples demonstrate the source coherence requirements for different display specifications following the analysis above. These examples start from the user-facing characteristics of the CGH display and use this to inform the choice of a suitable illumination source (or how the illumination source should be filtered for acceptable performance).

Example 1

[0077] In a first example, a CGH display has the following properties: [0078] The SLM is reimaged at 0.5 m from a viewer; [0079] Virtual content is presented in the range 0.25 m to infinity (+2 dioptres), giving D.sub.max=2 dioptres. [0080] The resolution, .sub.resolution, is 0.4 mrad (44 ppd/points per degree). [0081] The eyebox width of 2.5 mm. [0082] The horizontal field of view is 0.4 rad (23 degrees)

[0083] Applying these to inequalities (3) and (4) above results in G80 m.Math.rad and / 0.08. In this case an LED is likely to be appropriate, but one with a relatively small emitting area should be sourced. For example, collimating a central 0.4 rad (full width) emission cone allows for an emitter width of 200 m. The spectral bandwidth requirement allows for broad LED emissions, although very spectrally broad LEDs based around fluorescent phosphors may need to be avoided. RGB LEDs are suitable candidates and can allow display of colour images through sequential display of component images. An example suitable RGB LED is the DISPLIX P3333, KRTBLSLPS1.32 commercially available from Osram Opto Semiconductors.

Example 2

[0084] In a second example, the CGH display has a smaller horizontal field of view and the eyebox is limited by the maximum size of a viewer's pupil. This has the following properties: [0085] The SLM is reimaged at infinity. [0086] Virtual content is displayed in the range 0.25 m to infinity, giving D.sub.max=4 dioptres [0087] the resolution, .sub.resolution, is set to typical human visual acuity, approx. 0.15 mrad (116 ppd). [0088] The eyebox width is 7 mm (limited by the maximum size of the viewer's pupil) [0089] The horizontal field of view is 0.17 rad (10 degrees).

[0090] Applying these to inequalities (3) and (4) above results in G6 m.Math.rad; / 0.005. In this case an LED would need to be filtered significantly to provide a low enough etendue and small enough spectral bandwidth, and the resulting efficiency is likely to be too low. However, a multimode laser could still be used, for example coupled into a 25 m, 0.1 NA multimode fiber.

Example 3

[0091] In a third example, a large horizontal field of view is required with a small eyebox and relatively low resolution. This has the following properties: [0092] The SLM is reimaged 1 m from a viewer. [0093] Virtual content is displayed in the range 0.5 m to infinity (+1 diopters), giving D.sub.max=1 diopters. [0094] The resolution, .sub.resolution, is about 0.6 mrad (29 ppd). [0095] The eyebox is 1 mm wide [0096] The horizontal field of view is 1.0 rad (57 degrees).

[0097] Applying these to inequalities (3) and (4) above results in G600 m.Math.rad; /0.6. In this case, an LED with a relatively large (1 mm) emitting area may be used, together with a low f condenser lens (or possibly no condenser lens) coupling a large emission angle. The spectral bandwidth requirement poses no practical limitation, and a broadband LED based on a fluorescent phosphor may be used.

Examples 4 to 7Designing a CGH Display for a Particular Illumination Source

[0098] The skilled person is able to design CGH display systems to give predefined user-facing specifications as defined in equations (3) and (4) above. This presents an alternative way to use this disclosure; to design a CGH display system starting from a chosen illumination source. For example, a particular illumination source may have a particular desired wavelength (or wavelengths), or it may be desired to increase efficiency by reducing a level of filtering without unduly reducing resolution.

Example 4Compact RGB LED

[0099] The Osram OSIRE E3323, KRTBDWLM32.32 commercially available from Osram Opto Semiconductors has individually controllable RGB dies in a compact package. Its form factor, luminous flux and power consumption are appropriate for an untethered Head Mounted Device (HMD) (approximately 1 lumen at approximately 100 mW). From the data sheet (English version 1.1 2020 Nov. 25) the following parameters are obtained: .sub.peak=635 nm, 526 nm, 456 nm; =20 nm; 31 nm; 26 nm (FWHM); approximately 0.25 mm0.25 mm emitter size (separate die per colour, each approximately this size); approximately a Lambertian emitter. It is assumed that a central 1 radian full-width cone is collimated for illumination of the SLM. This gives G=250 m.Math.rad and /=0.031; 0.059; 0.057 (R;G;B)

[0100] It is now possible set the other parameters of the system to fit this source through inequalities (3) and (4) above. For example: [0101] D.sub.max1.5 dioptres (SLM reimaged at 67 cm from eye; content displayed between 33 cm and infinity) [0102] FoV50 degrees [0103] Resolution40 ppd (points per degree); .sub.resolution0.4 mrad [0104] Eyebox5 mm

Example 5High Power, High Bandwidth LED

[0105] The M565D2 commercially available from Thorlabs is a high brightness, high power green LED. It likely uses a fluorescent emitter (a UV LED emitting into green phosphor) to achieve high power with the desired colour, resulting in a very broad spectral bandwidth. This is suitable for a tethered Head Mounted Display (HMD) requiring high brightness over a large FoV and limited focal depth (for example an aerospace HMD). From the datasheet (Sep. 27, 2019, MTN003919-S01, Rev D) the following information is obtained: =565 nm; =104 nm (FWHM); 1 mm1 mm emitter size (assumed to be an approximately Lambertian emitter)

[0106] As with example 4, we assume a central 1 radian full-width cone is collimated for illumination of SLM. This gives G=1000 m.Math.rad; /=0.18

[0107] It is now possible set the other parameters of the system to fit this source through inequalities (3) and (4) above. For example: [0108] D.sub.max1 dioptre (SLM reimaged at 1 m from eye; content displayed between 50 cm and infinity) [0109] FoV100 degrees [0110] Resolution30 ppd; .sub.resolution0.6 mrad [0111] Eyebox3 mm

Example 6Multimode Fiber-Pigtailed Laser Diode

[0112] The PL52E0252FCB-T commercially available from MKS Newport is likely formed from a single transverse mode emitter which has been coupled into a multimode fiber for improved efficiency (compared to coupling into a single mode fiber). In a CGH display, this is likely to be used with a de-speckler to act as a more uniform multimode source. In this example, the spectral bandwidth is assumed not to be a limitation (as the source is a laser so it can be assumed to be small to satisfy equation (2). However, equation (1) allows the relatively small source etendue to be used for high resolution even into a small Field of View (FoV). This may be useful to provide a high-quality display which occupies a small part of the overall field of view of a HMD, in much the same way that a watchor smart watchcan provide useful information while only forming small part of the FoV. It might provide a display of time, status or other data. The status or other data could include exercise data, such as heart rate and number of steps, and motion data, such as current and average speed. From the datasheet (DS-072002_08/20) we learn that =520 nm; =1 nm (not specified but assumed to be typical for this type of laser); 50 m 0.2 NA fiber. This gives G=20 m.Math.rad; /=0.002

[0113] It is now possible set the other parameters of the system to fit this source through inequalities (3) and (4) above. For example: [0114] D.sub.max2 dioptres (SLM reimaged at 50 cm from eye; content displayed between 25 cm and infinity) [0115] FoV10 degrees [0116] Resolution80 ppd; .sub.resolution0.2 mrad [0117] EyeboxN/A (no limitation imposed by spectral bandwidth)

Example 7Superluminescent LED (SLED or SLD)

[0118] The EXS210115-00, commercially available from Exalos AG Switzerland is a single Transverse Mode source with broad spectral emission (compared to a laser). As with the laser of example 6, it could be coupled into a multimode fiber, but the broad spectral bandwidth would avoid the requirement for additional de-speckling. From the Exalos website (https://www.exalos.com/sled-modules/) it has the following properties: =510 nm; =10 nm; single transverse mode. This gives G==0.51 m.Math.rad (single transverse mode); /=0.02

[0119] It is now possible set the other parameters of the system to fit this source through equations (1) and (2) above. For example: [0120] D.sub.max1.5 dioptre (SLM reimaged at 67 cm from eye; content displayed between 33 cm and infinity) [0121] FoVN/A (no limitation due to single transverse mode) [0122] Resolution80 ppd; .sub.resolution0.2 mrad [0123] Eyebox7 mm (i.e., to match largest expected pupil size)

Example ComputerGenerated Hologram Display System

[0124] An example computer-generated hologram (CGH) display system that can be used with the principles of this disclosure is depicted in diagrammatic form in FIG. 3.

[0125] The CGH display system comprises an illumination system 300, and an SLM 306. The illumination system 300 comprises an illumination source 302 and illumination optics 304, in this case comprising a collimating lens.

[0126] Light from the illumination source 302 passes through the illumination optics 304 before illuminating a spatial light modulator (SLM) 306. Light incident on the SLM has an etendue G, spectral bandwidth and nominal wavelength . After the SLM 306, light forms an image plane 310, some distance from a viewing position of a viewer's eye 312. A characteristic of such a display system is that the SLM is re-imaged into an image plane 310 a distance from a viewer's pupil, and it is a maximum focal power of a virtual point displayed relative to the image plane 310 that sets a limit on the acceptable properties of the illumination system.

[0127] FIG. 3 is a simplified representation, and further examples may have additional components, such as a relay lens discussed in more detail below with reference to FIG. 6. Although FIG. 3 depicts a transmissive SLM for clarity, the present disclosure applies equally to reflective SLMs. Furthermore, the skilled person will appreciate that the present disclosure can be applied generally to CGH displays that reimage the SLM a distance from the viewing position, and is not limited to the particular form of FIG. 3.

Relationship to SLM Parameters

[0128] The discussion so far has been independent of the SLM. Often, but not necessarily, the target resolution will be related to the field of view or eyebox size. If the resolution of a display matches the pixel size of the SLM then we can write:

[00012]${}_{\mathrm{resolution}}={}_{\mathrm{fov}}/n$

where n is the number of pixels across an SLM. Equating .sub.resolution with .sub.spatial and substituting into equation (1) we can then write:

[00013]$Gn{}_{\mathrm{resolution}}^2/D_{\max }$

[0129] Additionally, If the SLM is relayed to the eye by a lens of focal length f.sub.r, then .sub.resolution=p/f.sub.r, where p is the pixel pitch of the SLM, and

[00014]$Gn{p}^2/\left(D_{\max }{f}_r^2\right)$

[0130] If the source is an extended source, width , collimated by a lens having focal length f.sub.c, then the angular extent of the source required to fill the SLM is given by np/f.sub.c. Substituting in G=np/f.sub.c gives us:

[00015]$/f_{c}p/\left(D_{\max }{f}_r^2\right)$

[0131] Assuming 1/f.sub.r>>D.sub.max, we can write d.sub.max=D.sub.maxf.sub.r.sup.2, where d.sub.max is the maximum distance from the SLM that an image point is formed, and hence write:

[00016]$\begin{array}{cc}{0}_{s}p/d_{\max }& \left(5\right)\end{array}$

where .sub.s=/f.sub.c is the angular subtense of the source at the SLM.

[0132] Similarly for temporal coherence, often the target resolution of a display is determined by the diffraction-limited resolution of a given eyebox size, according to the relationship:

[00017]${}_{\mathrm{resolution}}=/w_{\mathrm{eyebox}}$

Equating .sub.resolution with .sub.spatial and substituting into (2) we can then write:

[00018]${}_{\mathrm{resolution}}^2/D_{\max }$

Again, using the substitutions .sub.resolution=p/f.sub.r and d.sub.max=D.sub.maxf.sub.r.sup.2 we can then write:

[00019]$\begin{array}{cc}{p}^2/d_{\max }& \left(6\right)\end{array}$

Further Discussion Considering SLM Properties

[0133] In the following sections, additional analysis considering contributions to a point spread function (PSF) of the display system is presented. From this, constraints on characteristics of an illumination system can be defined qualitatively to assist choice of illumination source within the illumination system. A width of the combined point spread function of the display system (w.sub.PSF) has contributions from the angular subtense of the source at the SLM (w.sub.spatial), the source's spectral bandwidth (w.sub.temporal), and the inherent SLM resolution (w.sub.SLM). These contributions to the width are additive according to the following relation:

[00020]$\begin{array}{cc}{w}_{\mathrm{PSF}}=\sqrt{w_{{\mathrm{SLM}}^2}+w_{{\mathrm{spatial}}^2}+w_{{\mathrm{temporal}}^2}}& \left(7\right)\end{array}$

[0134] In the display systems discussed here, the illumination system is designed so that neither of the coherence-related factors (w.sub.spatial and w.sub.temporal) are the dominant effect on resolution as represented by w.sub.PSF, so the following constraints are used:

[00021]$\begin{array}{cc}{w}_{\mathrm{spatial}}{w}_{\mathrm{SLM}}& \left(8\right)\end{array}$$w_{\mathrm{temporal}}{w}_{\mathrm{SLM}}$

[0135] The inherent SLM resolution, w.sub.SLM, depends on the nature of the display system. Where the full extent of the SLM is used directly, w.sub.SLM can be approximated to a pixel pitch of the SLM, p. In other cases, spatial filtering after the SLM can reduce inherent resolution, for example to twice the pixel pitch, 2p. From the inequalities in (8), a larger value of w.sub.SLM means that the contribution from the illumination system characteristics can be greater. It will be appreciated that although two constraints may be applied to the illumination system, other examples may apply other numbers of constraints, such as a single one.

Transverse Coherence Width (w.sub.spatial)

[0136] Referring to FIG. 4, the elements of the display system design that contribute to w.sub.spatial are illustrated. An illumination system 402, has a source diameter and a numerical aperture NA. (The numerical aperture is a measure of the divergence of the source). Light from the illumination system 402 passes through a collimating lens 404 having a focal length f.sub.c to substantially collimate the light when it is incident on the SLM 406. As can be seen in FIG. 4, the collimating lens may not lead to light being exactly parallel, it might be slightly converging (as shown) or slightly diverging, so that the light from the source substantially illuminates the SLM with minimal wasted light outside the SLM, to improve efficiency.

[0137] Light incident on the SLM 406 has an angular subtense s which is not necessarily the same as the numerical aperture, NA, of the source 402. Using the small angle approximation,

[00022]${}_s\frac{}{f_c}.$

In general, at a distance d from the SLM, the transverse coherence width is related to the angular subtense of the source at the display s and the displayed contents' axial distance from the SLM, d:

[00023]$\begin{array}{cc}{w}_{\mathrm{spatial}}d{}_s\frac{d}{f_c}& \left(9\right)\end{array}$

Temporal Coherence Width (w.sub.temporal)

[0138] Referring to FIG. 5, the elements of the display system design that contribute to the temporal coherence width, w.sub.temporal, are illustrated. Light leaving the SLM 506 is limited in terms of how much it can be brought into a singular focus by the spectral band of the illumination system.

[0139] The temporal coherence of the illumination system determines the degree to which a spectral band from the source (, centred on ) can be brought to a singular focus at an axial distance from the SLM(d).

[0140] This is bounded by the maximum diffraction angle achievable from the SLM () and pixel pitch (p) by the standard equation for a blazed grating:

[00024]$\begin{array}{cc}p\left(\sin +\sin \right)=m& \left(10\right)\end{array}$

where is the incidence angle, which can be assumed to be zero because light incident on the SLM is generally collimated, and m is the diffraction order, taken as 1 in this case. From this, and applying the small angle approximation, it follows that:

[00025]$\begin{array}{cc}={\sin }^{-1}\left(\frac{}{p}\right)\frac{}{p}& \left(11\right)\end{array}$

[0141] This the informs the value of w.sub.temporal:

[00026]$\begin{array}{cc}{w}_{\mathrm{temporal}}\left(\frac{d-d^{}}{d^{}}\right)& \left(12\right)\end{array}$$d=\frac{d}{p}$$\left(\mathrm{where}\frac{d^{}}{d}=\frac{}{{}^{}},{}^{}=+\right)$

here is the wavelength corresponding to d, and d represents the out-of-focus components of in the bundle of wavelengths.

Relay Lens

[0142] FIG. 6 is a diagrammatic action of a relay lens. An output relay lens 608 (with focal length f.sub.r) maps object space to image space, which scales axial distance from the SLM 606. Some embodiments may omit the relay lens but, when it is included, it can be useful to ensure that content (imaged at a real distance d.sub.max from the SLM) is not placed within the viewer's near point (z.sub.min, the nearest point that an eye can bring into focus, typically about 25 cm and closer). The relay lens allows a limit to be placed on d.sub.max for the design of the system but the characteristics of the lens itself cancel out from the calculations. Nonetheless, the relay lens allows d.sub.max to be made smaller, which is beneficial for the system because a smaller value of d.sub.max allows wider spectral bandwidth and angular subtense in the light leaving the SLM, so that a broader variety of possible light sources can be used.

[00027]$\begin{array}{cc}\frac{1}{f_r}=\frac{1}{u}-\frac{1}{z_{\min }}.Math.u=\frac{f_r{z}_{\min }}{f_r+z_{\min }}& \left(13\right)\end{array}$$d_{\max }=f_r-u=f_r-\frac{f_r{z}_{\min }}{f_r+z_{\min }}=\frac{f_r^2}{f_r+z_{\min }}$$f_r\frac{\mathrm{np}}{{}_{\mathrm{FOV}}}$

where f.sub.r is the focal length of the relay lens, z.sub.min is a minimum focal distance of a viewer, u is the distance from the viewer to the reimaged SLM, n is the number of pixels/points in a given axis (for example, 1920 in the horizonal axis for a 1080p display) and .sub.FOV is the field of view of the viewer.

System Design Using SLM Properties

[0143] We can now determine the constraints on the system as follows

[00028]$\mathrm{Spatial}\mathrm{coherence}\mathrm{limit}-$$\begin{array}{cc}\frac{d_{\max }}{f_c}p.Math.\frac{p{f}_c}{d_{\max }}\mathrm{and}/\mathrm{or}{}_s\frac{p}{d_{\max }}& \left(14\right)\end{array}$$\mathrm{Temporal}\mathrm{coherence}\mathrm{limit}-$$\begin{array}{cc}\frac{d_{\max }}{p}p.Math.\frac{p^2}{d_{\max }}& \left(15\right)\end{array}$

[0144] Using this set of equations, it is now possible to determine quantitively characteristics of the illumination system spectral bandwidth, diameter, and/or angular subtense in a way not previously possible with reference to the properties of the CGH display system and the SLM. Suitable illumination sources can be selected based on cost, brightness, efficiency or other factors with knowledge that acceptable image sharpness will be achieved. While it remains the case that not all LEDs, for example, are suitable, it is possible to select suitable LEDs for a particular system design with confidence.

Method of Defining a Suitable Light Source when SLM Properties are Considered

[0145] FIG. 7 depicts a method of selecting a suitable light source for a computer-generated hologram display system according to an example using the analysis considering SLM properties discussed above. Although the blocks in FIG. 7 are depicted sequentially, the method is not limited to this order and block may be carried out in a different order and/or in parallel in other examples.

[0146] First, the optical system is designed, or parameters of an existing CGH display system are gathered. At 702, details of a spatial light modulator (SLM) having a pixel pitch is specified or gathered from the datasheet of the SLM. At 704 a predetermined maximum distance at which the SLM will be re-imaged in use is specified. Other characteristics of the design may also be specified, such as a collimating lens focal length and/or a focal length of a relay lens, if included.

[0147] With the system design known, at 706 a contribution to a point spread function (PSF) by the SLM at the predetermined maximum distance is determined. For example, this may simply be the pixel pitch assuming that incident light is substantially collimated when illuminating the SLM and so generally remains substantially collimated after leaving the SLM. This contribution to a PSF is used as a constraint to determine a maximum spectral bandwidth of the illumination system based on the predetermined maximum distance at 708, such as by applying equation (15) above. A maximum angular subtense leaving the SLM is determined at 710 based on the predetermined maximum distance and using the contribution to the point spread function by the SLM at the predetermined maximum distance as a constraint, for example by applying equation (14) above for s. At 712, a maximum diameter of the light source is determined, such as by applying equation 8 above for . Some examples may use one, two or all three of the determinations at 708, 710 and 712, the more that are used the tighter the criteria for the light source and the higher the image quality. Specifying angular subtense s and may be useful when the angular subtense after the SLM is not the same as the numerical aperture of the source.

[0148] Finally, at 714, a light source is selected for an illumination system based on the determined characteristics.

[0149] It can be understood that the method of FIG. 7 can be used to qualitatively inform selection of a light source for good image quality. In other examples, the design of a CGH display system may itself be influenced, such as to allow a wider choice of light sources, or to allow a particular light source to be used. For example, it might be desired to use a Superluminescent LED to achieve a high luminance, and knowledge of the spectral bandwidth could influence the other variables of the display system, such as determining a value for d.sub.max that allows sufficient spectral bandwidth for the source to be used.

[0150] Other design methods may be used in other examples. For example, the design of CGH display systems in Examples 1 to 7 above started with user facing properties of the CGH system independent of the SLM, or properties of the illumination system in terms of etendue and/or spectral bandwidth.

[0151] The above embodiments are to be understood as illustrative examples of the invention. Further embodiments of the invention are envisaged. For example, while the system includes a collimating lens, this may not be required if the numerical aperture of the source is sufficiently small. It is to be understood that any feature described in relation to any one embodiment may be used alone, or in combination with other features described, and may also be used in combination with one or more features of any other of the embodiments, or any combination of any other of the embodiments. Furthermore, equivalents and modifications not described above may also be employed without departing from the scope of the invention, which is defined in the accompanying claims.